3 Answers
3

Update: Saving in multiplayer is now simply an item on the menu, accessible by pressing escape. (I can't recall off the top of my head if only the host can save, or if any player can save.)

Old answer from before saving was simple below:

No, you can not manually save in multiplayer. You can however change your settings so that Civilization 5 autosaves your game every turn.

Now to see if I can find some links to support my answer...

EDIT: The Civilization 5 manual claims the that HOST can save, and if the host disconnects someone else is nominated host. Having never hosted, I can't confirm if this is possible. I believe every player can still autosave as usual.

When hosting a game, you can save your game’s progress and load it up at a later date, especially useful for long-running games which can span longer than a day. From within
the Setup Multiplayer Game Screen, click the “Load Game” button to browse your list of saved games.

If for any reason you become disconnected while playing a Multiplayer game, the system will try to reconnect you automatically. If the host of a game leaves (for whatever reason), the game will attempt to migrate the host controls to a different
player.

There's no menu item for it, but you can bring up the save dialog by pressing Ctrl+S.

If you want to reload your save (which would make sense I suppose), you will need to dupe the game into thinking it's an autosave by moving it from the My Games\Sid Meier's Civilization 5\Saves\multi directory to the auto directory one level below that and prefixing the filename with "AutoSave_".

I'm sure this functionality will be included in a patch in the near future.

Alternatively, you can just rely on the actual autosaves which are created for every player.